Making robust predictions for the phase-space distribution of dark matter at the solar neighbourhood is vital for dark matter direct-detection experiments. To date, almost all such predictions have been based on simulations that model the dark matter alone. Here, we use three cosmological hydrodynamic simulations of bright, disc-dominated galaxies to include the effects of baryonic matter self-consistently for the first time. We find that the addition of baryonic physics drastically alters the dark matter profile in the vicinity of the solar neighbourhood. A stellar/gas disc, already in place at high redshift, causes merging satellites to be dragged preferentially towards the disc plane where they are torn apart by tides. This results in an accreted dark matter disc that contributes ∼0.25–1.5 times the non-rotating halo density at the solar position. The dark disc, unlike dark matter streams, is an equilibrium structure that must exist in disc galaxies that form in a hierarchical cosmology. Its low rotation lag with respect to the Earth significantly boosts Weakly Interacting Massive Particle (WIMP) capture in the Earth and Sun, boosts the annual modulation signal and leads to distinct variations in the flux as a function of recoil energy that allow the WIMP mass to be determined.